**PLANE CURVES AND FREE HAND SKETCHING**

**CONSTRUCTION OF PARALLEL & PERPENDICULAR LINES**

**1.Given are the steps to draw a perpendicular**

to a line at a point within the line when the

point is near the Centre of a line.

Arrange the steps. Let AB be the line and P

be the point in it

i. P as Centre, take convenient radius R1 and

draw arcs on the two sides of P on the line at

C, D.

ii. Join E and P

iii. The line EP is perpendicular to AB

iv. Then from C, D as Centre, take R2 radius

(greater than R1), draw arcs which cut at E.

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, i, iv, iii**Answer: a**

**Explanation:** Here uses the concept of a locus. Every 2 points have a particular line

that is every point on line is equidistant from

both the points. The above procedure shows

how the line is build up using arcs of the

similar radius.

**2.Given are the steps to draw a perpendicular**

to a line at a point within the line when the

point is near an end of the line.

Arrange the steps. Let AB be the line and P

be the point in it.

i. Join the D and P.

ii. With any point O draw an arc (more than a

semicircle) with a radius of OP, cuts AB at C.

iii. Join the C and O and extend till it cuts the

large arc at D.

iv. DP gives the perpendicular to AB.

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, iii, i, iv**Answer: d****Explanation:**There exists a common

procedure for obtaining perpendiculars for

lines. But changes are due changes in

conditions whether the point lies on the line,

off the line, near the centre or near the ends

etc.**3.Given are the steps to draw a perpendicular**

to a line at a point within the line when the

point is near the centre of line.

Arrange the steps. Let AB be the line and P

be the point in it

i. Join F and P which is perpendicular to AB.

ii. Now C as centre take the same radius and

cut the arc at D and again D as centre with

same radius cut the arc further at E.

iii. With centre as P take any radius and draw

an arc (more than a semicircle) cuts AB at C.

iv. Now D, E as centre take radius (more than

half of DE) draw arcs which cut at F.

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, i, iv, iii

**Answer: b****Explanation:** Generally in drawing

perpendiculars to lines involves in drawing a

line which gives equidistance from either side

of the line to the base, which is called the

locus of points. But here since the point P is

nearer to end, there exists some peculiar steps

in drawing arcs.

**4.Given are the steps to draw a perpendicular**

to a line from a point outside the line, when

the point is near the centre of line.

Arrange the steps. Let AB be the line and P

be the point outside the line

i. The line EP is perpendicular to AB

ii. From P take convenient radius and draw

arcs which cut AB at two places, say C, D.

iii. Join E and P.

iv. Now from centers C, D draw arc with

radius (more than half of CD), which cut each

other at E.

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, iv, iii, i**Answer: d****Explanation:**At first two points are taken

from the line to which perpendicular is to

draw with respect to P. Then from two points

equidistant arcs are drawn to meet at some

point which is always on the perpendicular.

So by joining that point and P gives

perpendicular.

**5.Given are the steps to draw a perpendicular**

to a line from a point outside the line, when

the point is near an end of the line.**Arrange the steps. Let AB be the line and P**

be the point outside the line

i. The line ED is perpendicular to AB

ii. Now take C as centre and CP as radius cut

the previous arc at two points say D, E.

iii. Join E and D.

iv. Take A as center and radius AP draw an

arc (semicircle), which cuts AB or extended

AB at C.

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, ii, i, iii

d) ii, iv, iii, i**Answer: c****Explanation:**The steps here show how to

draw a perpendicular to a line from a point

when the point is nearer to end of line. Easily

by drawing arcs which are equidistance from

either sides of line and coinciding with point

P perpendicular has drawn.**6.Given are the steps to draw a perpendicular**

to a line from a point outside the line, when

the point is nearer the centre of line.

Arrange the steps. Let AB be the line and P

be the point outside the line

i. Take P as centre and take some convenient

radius draw arcs which cut AB at C, D.

ii. Join E, F and extend it, which is

perpendicular to AB.

iii. From C, D with radius R1 (more than half

of CD), draw arcs which cut each other at E.

iv. Again from C, D with radius R2 (more

than R1), draw arcs which cut each other at F.

a) i, iii, iv, ii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, iv, iii, i**Answer: a****Explanation:**For every two points there

exists a line which has points from which

both the points are equidistant otherwise

called perpendicular to line joining the two

points. Here at 1st step, we created two on the

line we needed perpendicular, then with equal

arcs from either sides we created the

perpendicular.

**7.Given are the steps to draw a parallel line**

to given line AB at given point P.

Arrange the steps.

i. Take P as centre draw a semicircle which

cuts AB at C with convenient radius.

ii. From C with radius of PD draw an arc with

cuts the semicircle at E.

iii. Join E and P which gives parallel line to AB. iv. From C with same radius cut the AB at D.

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, iv, iii, i**Answer: a**

Explanation: There exists some typical steps

in obtaining parallel lines for required lines at

given points which involves drawing of arcs,

necessarily, here to form a parallelogram

since the opposite sides in parallelogram are

parallel.

**8.Given are the steps to draw a parallel line**

to given line AB at a distance R.

Arrange the steps.

i. EF is the required parallel line.

ii. From C, D with radius R, draw arcs on the

same side of AB.

iii. Take two points say C, D on AB as far as

possible.

iv. Draw a line EF which touches both the arc

(tangents) at E, F.

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, iv, iii, i**Answer: b****Explanation:**Since there is no reference

point P to draw parallel line, but given the

distance, we can just take arcs with distance

given from the base line and draw tangent

which touches both arcs.**9.Perpendiculars can’t be drawn using**

a) T- Square

b) Set-squares

c) Pro- circle

d) Protractor**Answer: c****Explanation:**T-square is meant for drawing a straight line and also perpendiculars. And also using set-squares we can draw perpendiculars. Protractor is used to measure angles and also we can use to draw perpendiculars. But pro-circle consists of circles of different diameters.

**10.The length through perpendicular gives**

the shortest length from a point to the line.

a) True

b) False**Answer: a****Explanation:**The statement given here is

right. If we need the shortest distance from a

point to the line, then drawing perpendicular

along the point to a line is the best method.

Since the perpendicular is the line which has

points equidistant from points either side of

given line.

**DRAWING REGULAR POLYGONS & SIMPLE CURVES**

**1.A Ogee curve is a**

a) semi ellipse

b) continuous double curve with convex and

concave

c) freehand curve which connects two parallel

lines

d) semi hyperbola**Answer: b****Explanation:**An ogee curve or a reverse

curve is a combination of two same curves in

which the second curve has a reverse shape to

that of the first curve. Any curve or line or

mould consists of a continuous double curve

with the upper part convex and lower part

concave, like ‘’S’’.

**2.Given are the steps to construct an**

equilateral triangle, when the length of side is

given. Using, T-square, set-squares only.

Arrange the steps.

i. The both 2 lines meet at C. ABC is required

triangle

ii. With a T-square, draw a line AB with

given length

iii. With 30o-60o set-squares, draw a line

making 60o with AB at A

iv. With 30o-60o set-squares, draw a line

making 60o with AB at B

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, iii, iv, i**Answer: d****Explanation:** Here gives the simple

procedure since T-square and 30o

-60o setsquares. And also required triangle is

equilateral triangle. The interior angles are

60°, 60°, 60° (180° /3 = 60°). Set- squares are

used for purpose of 60°.

**3.Given are the steps to construct an**

equilateral triangle, with help of a compass,

when the length of a side is given. Arrange

the steps.

i. Draw a line AB with given length

ii. Draw lines joining C with A and B

iii. ABC is required equilateral triangle

iv. With centers A and B and radius equal to

AB, draw arcs cutting each other at C

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, iii, iv, i**Answer: a****Explanation:**Here gives the simple

procedure to construct an equilateral triangle.

Since we used compass we can construct any

type of triangle but with set-squares it is not

possible to construct any type of triangles

such as isosceles, scalene etc.**4.Given are the steps to construct an**

equilateral triangle when the altitude of a

triangle is given. Using, T-square, set-squares

only. Arrange the steps.

i. Join R, Q; T, Q. Q, R, T is the required

triangle

ii. With a T-square, draw a line AB of any

length

iii. From a point P on AB draw a

perpendicular PQ of given altitude length

iv. With 30o-60o set-squares, draw a line

making 30o with PQ at Q on both sides

cutting at R, T

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, iii, iv, i**Answer: d****Explanation:**Here gives the simple

procedure since T-square and 30°-60°setsquares. The interior angles are 60°, 60°, 60°

(180° /3 = 60°). Altitude divides the sides of equilateral triangle equally. Set- squares are

used for purpose of 30°.**5.Given are the steps to construct an**

equilateral triangle, with help of a compass,

when the length of altitude is given. Arrange

the steps.

i. Draw a line AB of any length. At any point

P on AB, draw a perpendicular PQ equal to

altitude length given

ii. Draw bisectors of CE and CF to intersect

AB at R and T respectively.QRT is required

triangle

iii. With center Q and any radius, draw an arc

intersecting PQ at C

iv. With center C and the same radius, draw

arcs cutting the 1st arc at E and F

a) i, iii, iv, ii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, iii, iv, i**Answer: a****Explanation:**This is the particular procedure

used for only constructing an equilateral

triangle using arcs when altitude is given

since we used similar radius arcs to get 30o

on both sides of a line. Here we also bisected

arc using the same procedure from bisecting

lines.

**6.How many pairs of parallel lines are there**

in regular Hexagon?

a) 2

b) 3

c) 6

d) 1**Answer: b****Explanation:**Hexagon is a closed figure

which has six sides, six corners. Given is

regular hexagon which means it has equal

interior angles and equal side lengths. So,

there will be 3 pair of parallel lines in a

regular hexagon.**7.Given are the steps to construct a square when the length of a side is given. Using, T square, set-squares only. Arrange the steps.**

i. Repeat the previous step and join A, B, C

and D to form a square

ii. With a T-square, draw a line AB with

given length.

iii. At A and B, draw verticals AE and BF

iv. With 45o set-squares, draw a line making

45o with AB at A cuts BF at C

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, iii, iv, i**Answer: d****Explanation:**Square is closed figure with

equal sides and equal interior angles which is

90°. In the above steps, it is given the

procedure to draw a square using set-squares.

45° set-square is used since 90/2 = 45.**8.How many pairs of parallel lines are there**

in a regular pentagon?

a) 0

b) 1

c) 2

d) 5**Answer: a****Explanation:**Pentagon is a closed figure

which has five sides, five corners. Given is

regular pentagon which means it has equal

interior angles and equal side lengths. Since

five is odd number so, there exists angles 36°,

72°, 108°, 144°, 180° with sides to

horizontal.**9.Given are the steps to construct a square**

using a compass when the length of the side

is given. Arrange the steps.

i. Join A, B, C and D to form a square

ii. At A with radius AB draw an arc, cut the

AE at D

iii. Draw a line AB with given length. At A

draw a perpendicular AE to AB using arcs

iv. With centers B and D and the same radius,

draw arcs intersecting at C

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, iii, iv, i**Answer: b****Explanation:**Here we just used simple

techniques like drawing perpendiculars using

arcs and then used the compass to locate the

fourth point. Using the compass it is easier to

draw different types of closed figures than

using set-squares.

**10.Given are the steps to construct regular**

polygon of any number of sides. Arrange the

steps.

i. Draw the perpendicular bisector of AB to

cut the line AP in 4 and the arc AP in 6

ii. The midpoint of 4 and 6 gives 5 and

extension of that line along the equidistant

points 7, 8, etc gives the centers for different

polygons with that number of sides and the

radius is AN (N is from 4, 5, 6, 7, so on to N)

iii. Join A and P. With center B and radius

AB, draw the quadrant AP

iv. Draw a line AB of given length. At B,

draw a line BP perpendicular and equal to AB

a) i, iv, ii, iii

b) iii, ii, iv, i

c) iv, iii, i, ii

d) ii, iii, iv, i**Answer: c****Explanation: **Given here is the method for

drawing regular polygons of a different

number of sides of any length. This includes

finding a line where all the centers for regular

polygons lies and then with radius taking any

end of 1st drawn line to center and then

completing circle at last, cutting the circle

with the same length of initial line. Thus we

acquire polygons.

**DRAWING TANGENTS AND NORMALS FOR DIFFERENT CONDITIONS OF CIRCLE**

**1.Given are the steps to draw a tangent to**

any given circle at any point P on it. Arrange

the steps.

i. Draw the given circle with center O and

mark the point P anywhere on the circle.

ii. With centers O and Q draw arcs with equal

radius to cut each other at R.

iii. Join R and P which is the required

tangent.

iv. Draw a line joining O and P. Extend the

line to Q such that OP = PQ.

a) i, iv, ii, iii

b) iv, i, iii, ii

c) iii, i, iv, ii

d) ii, iv, i, iii**Answer: a**Tangent is a line which touches

Explanation:

a curve at only one point. Every tangent is

perpendicular to its normal. Here we first

found the normal which passes through center

and point. Then drawing a perpendicular to it

gives the tangent.**2.Given are the steps to draw a tangent to**

given circle from any point outside the circle.

Arrange the steps.

i. With OP as diameter, draw arcs on circle at

R and R1.

ii. Draw the given circle with center O.

iii. Join P and R which is one tangent and

PR1 is another tangent.

iv. Mark the point P outside the circle.

a) ii, iv, iii, i

b) iv, i, iii, ii

c) iii, i, iv, ii

d) ii, iv, i, iii**Answer: d**Usually when a point is outside

Explanation:

the circle there exists two tangents. For which

we first join the center with point P and then

taking distance from center to P as diameter

circle is drawn from the midpoint of center

and P to cut circle at two points where

tangents touch the circle.**3.Given are the steps to draw a tangent to**

given arc even if center is unknown and the

point P lies on it. Arrange the steps. Let AB

be the arc.

i. Draw EF, the bisector of the arc CD. It will

pass through P.

ii. RS is the required tangent.

iii. With P as center and any radius draw arcs

cutting arc AB at C and D.

iv. Draw a perpendicular RS to EF through P.

a) ii, iv, iii, i

b) iv, i, iii, ii

c) iii, i, iv, ii

d) ii, iv, i, iii**Answer: c**Even if the center of the arc is

Explanation:

unknown, just by taking any some part of arc

and bisecting that with a line at required point

p gives us normal to tangent at P. So then

from normal drawing perpendicular gives our

required tangent.**4.Given are the steps to draw a tangent to**

given circle and parallel to given line.

Arrange the steps.

i. Draw a perpendicular to given line and

extend to cut the circle at two points P and Q

ii. At P or Q draw perpendicular to normal

then we get the tangents.

iii. PQ is the normal for required tangent.

iv. Draw a circle with center O and line AB as

required.

a) ii, iv, iii, i

b) iv, i, iii, ii

c) iii, i, iv, ii

d) ii, iv, i, iii**Answer: bExplanation: **Normal of curve will be

perpendicular to every parallel tangent at that

point. We just drawn the longest chord

(diameter) and then perpendicular it gives the

required tangents. Since circle is closed figure

there exist two tangents parallel to each other.

**5.How many external tangents are there for**

two circles?

a) 1

b) 2

c) 3

d) 4**Answer: b**External tangents are those

Explanation:

which touch both the circles but they will not

intersect in between the circles. The tangents

touch at outmost points of circles that are

ends of diameter if the circles have the same

diameter.**6.How many internal tangents are there for**

two circles?

a) 4

b) 3

c) 2

d) 1**Answer: c**Internal tangents are those

Explanation:

which touch both the circle and also intersect

each other on the line joining the centers of

circles. And the internal tangents intersect

each other at midpoint of line joining the

center of circles only if circles have the same

diameter.

**7.For any point on any curve there exist two**

normals.

a) True

b) False**Answer: b**Here we take point on the

Explanation:

curve. There exist multiple tangents for some

curve which are continuous, trigonometric

curves, hyperbola etc. But for curves like

circles, parabola, ellipse, cycloid etc. have

only one tangent and normal.**8.Arrange the steps. These give procedure to**

draw internal tangent to two given circles of

equal radii.

i. Draw a line AB which is the required

tangent.

ii. Draw the given circles with centers O and

P.

iii. With center R and radius RA, draw an arc

to intersect the other circle on the other circle

on the other side of OP at B.

iv. Bisect OP in R. Draw a semi circle with

OR as diameter to cut the circle at A.

a) ii, iv, iii, i

b) iv, i, iii, ii

c) iii, i, iv, ii

d) ii, iv, i, iii**Answer: a**Since the circles have same

Explanation:

radius. The only two internal tangents will

intersect at midpoint of line joining the

centers. So we first found the center and then

point of intersection of tangent and circle then

from that point to next point it is drawn a arc

midpoint as center and join the points gave us

tangent.

**9.There are 2 circles say A, B. A has 20 units**

radius and B has 10 units radius and distance

from centers of A and B is 40 units. Where

will be the intersection point of external

tangents?

a) to the left of two circles

b) to the right of the two circles

c) middle of the two circles

d) they intersect at midpoint of line joining

the centers**Answer: b**A has 20 units radius and B has

Explanation:

10 units radius. So, the tangents go along the

circles and meet at after the second circle that

is B that is the right side of both circles. And

we asked for external tangents so they meet

away from the circles but not in between

them.

10.**10.There are 2 circles say A, B. A is smallerthan B and they are not intersecting at anypoint. Where will be the intersection point ofinternal tangents for these circles?**

a) to the left of two circles

b) to the right of the two circles

c) middle of the two circles

d) they intersect at midpoint of line joining

the centers

**Answer: b**

Explanation:A is smaller than B so the

Explanation:

intersection point of internal tangents will not

be on the midpoint of the line joining the

centers. And we asked for internal tangents so

they will not meet away from the circles.

They meet in between them

**CONSTRUCTION OF ELLIPSE – 1**

**1.Which of the following is incorrect about**

Ellipse?

a) Eccentricity is less than 1

b) Mathematical equation is X^{2}/a^{2}+ Y^{2}/b^{2}=1

c) If a plane is parallel to axis of cone cuts the

cone then the section gives ellipse

d) The sum of the distances from two focuses

and any point on the ellipse is constant

Answer: c

Explanation: If a plane is parallel to the axis

of cone cuts the cone then the cross-section

gives hyperbola. If the plane is parallel to

base it gives circle. If the plane is inclined

with an angle more than the external angle of

cone it gives parabola. If the plane is inclined

and cut every generators then it forms an

ellipse.**2.Which of the following constructions**

doesn’t use elliptical curves?

a) Cooling towers

b) Dams

c) Bridges

d) Man-holes**Answer: a**Cooling towers, water channels use Hyperbolic curves as their design, Arches, Bridges, sound reflectors, lighter flectors etc use parabolic curves. Arches, bridges, dams, monuments, man-holes, glandsbridges, dams, monuments, man-holes, glands and stuffing boxes etc use elliptical curves.

Explanation:

**3.The line which passes through the focus**

and perpendicular to the major axis is

a) Minor axis

b) Latus rectum

c) Directrix

d) Tangent**Answer: b**The line bisecting the major

Explanation:

axis at right angles and terminated by curve is

called the minor axis. The line which passes

through the focus and perpendicular to the

major axis is latus rectum. Tangent is the line

which touches the curve at only one point.**4.Which of the following is the eccentricity**

for an ellipse?

a) 1

b) 3/2

c) 2/3

d) 5/2**Answer: c**The eccentricity for ellipse is

Explanation:

always less than 1. The eccentricity is always

1 for any parabola. The eccentricity is always

0 for a circle. The eccentricity for a hyperbola

is always greater than 1.**5.Axes are called conjugate axes when they**

are parallel to the tangents drawn at their

extremes.

a) True

b) False**Answer: a**In ellipse there exist two axes

Explanation:

(major and minor) which are perpendicular to

each other, whose extremes have tangents

parallel them. There exist two conjugate axes

for ellipse and 1 for parabola and hyperbola.**6.Steps are given to draw an ellipse by loop**

of the thread method. Arrange the steps.

i. Check whether the length of the thread is

enough to touch the end of minor axis.

ii. Draw two axes AB and CD intersecting at

O. Locate the foci F1 and F2.

iii. Move the pencil around the foci,

maintaining an even tension in the thread

throughout and obtain the ellipse.

iv. Insert a pin at each focus-point and tie a

piece of thread in the form of a loop around

the pins.

a) i, ii, iii, iv

b) ii, iv, i, iii

c) iii, iv, i, ii

d) iv, i, ii, iii**Answer: b**This is the easiest method of

Explanation:

drawing ellipse if we know the distance

between the foci and minor axis, major axis.

It is possible since ellipse can be traced by a

point, moving in the same plane as and in

such a way that the sum of its distances from

two foci is always the same.**7.Steps are given to draw an ellipse by**

trammel method. Arrange the steps.

i. Place the trammel so that R is on the minor

axis CD and Q on the major axis AB. Then P

will be on the ellipse.

ii. Draw two axes AB and CD intersecting

each other at O.

iii. By moving the trammel to new positions,

always keeping R on CD and Q on AB,

obtain other points and join those to get an

ellipse.

iv. Along the edge of a strip of paper which

may be used as a trammel, mark PQ equal to

half the minor axis and PR equal to half of

major axis.

a) i, ii, iii, iv

b) ii, iv, i, iii

c) iii, iv, i, ii

d) iv, i, ii, iii**Answer: b**This method uses the trammels PQ and PR which ends Q and R should be placed on major axis and minor axis respectively. It is possible since ellipse can be traced by a point, moving in the same plane as and in such a way that the sum of its distances from two foci is always the same.

Explanation:

**8.Steps are given to draw a normal and a**

tangent to the ellipse at a point Q on it.

Arrange the steps.

i. Draw a line ST through Q and

perpendicular to NM.

ii. ST is the required tangent.

iii. Join Q with the foci F1 and F2.

iv. Draw a line NM bisecting the angle

between the lines drawn before which is

normal.

a) i, ii, iii, iv

b) ii, iv, i, iii

c) iii, iv, i, ii

d) iv, i, ii, iii**Answer: c**Tangents are the lines which

Explanation:

touch the curves at only one point. Normals

are perpendiculars of tangents. As in the

circles first, we found the normal using foci

(centre in circle) and then perpendicular at

given point gives tangent.**9.Which of the following is not belonged to**

ellipse?

a) Latus rectum

b) Directrix

c) Major axis

d) Asymptotes**Answer: d**Latus rectum is the line joining

Explanation:

one of the foci and perpendicular to the major

axis. Asymptotes are the tangents which meet

the hyperbola at infinite distance. Major axis

consists of foci and perpendicular to the

minor axis.

**CONSTRUCTION OF ELLIPSE – 2**

**1.Mathematically, what is the equation of**

ellipse?

a) x^{2}/a^{2}+ y^{2}/b^{2}= -1

b) x^{2}/a^{2}– y^{2}/b^{2}= 1

c) x^{2}/a^{2}+ y^{2}/b^{2}= 1

d) x^{2}/a^{2}– y^{2}/b^{2}= 1**Answer: c**Equation of ellipse is given by;

Explanation:

x^{2}/a^{2}+ y^{2}/b^{2}= 1. Here, a and b are half the

distance of lengths of major and minor axes

of the ellipse. If the value of a = b then the

resulting ellipse will be a circle with Centre

(0,0) and radius equal to a units.**2.In general method of drawing an ellipse, a**

vertical line called as is drawn first.

a) Tangent

b) Normal

c) Major axis

d) Directrix**Answer: d**In the general method of

Explanation:

drawing an ellipse, a vertical line called as

directrix is drawn first. The focus is drawn at

a given distance from the directrix drawn.

The eccentricity of the ellipse is less than one.**3.If eccentricity of ellipse is 3/7, how many**

divisions will the line joining the directrix

and the focus have in general method?

a) 10

b) 7

c) 3

d) 5**Answer: a**: In the general method of

Explanation

drawing an ellipse, if eccentricity of the

ellipse is given as 3/7 then the line joining the

directrix and the focus will have 10 divisions.

The number is derived by adding the

numerator and denominator of the

eccentricity

**4.In the general method of drawing an**

ellipse, after parting the line joining the

directrix and the focus, a is made.

a) Tangent

b) Vertex

c) Perpendicular bisector

d) Normal**Answer: b**In the general method of

Explanation:

drawing after parting the line joining the

directrix and the focus, a vertex is made. An

arc with a radius equal to the length between

the vertex and the focus is drawn with the

vertex as the centre.**5.An ellipse is defined as a curve traced by a**

point which has the sum of distances between

any two fixed points always same in the same

plane.

a) True

b) False**Answer: a**An ellipse can also be defined

Explanation:

as a curve that can be traced by a point

moving in the same plane with the sum of the

distances between any two fixed points

always same. The two fixed points are called

as a focus.**6.An ellipse has foci.**

a) 1

b) 2

c) 3

d) 4**Answer: b**An ellipse has 2 foci. These

Explanation:

foci are fixed in a plane. The sum of the

distances of a point with the foci is always

same. The ellipse can also be defined as the

curved traced by the points which exhibit this

property.**7.If information about the major and minor**

axes of ellipse is given then by how many

methods can we draw the ellipse?

a) 2

b) 3

c) 4

d) 5**Answer: d**There are 5 methods by which

Explanation:

we can draw an ellipse if we know the major

and minor axes of that ellipse. Those five

methods are arcs of circles method,

concentric circles method, loop of the thread

method, oblong method, trammel method.**8.In arcs of circles method, the foci are**

constructed by drawing arcs with centre as

one of the ends of the axis and the

radius equal to the half of the axis.

a) Minor, major

b) Major, major

c) Minor, minor

d) Major, minor**Answer: a**In arcs of circles method, the

Explanation:

foci are constructed by drawing arcs with

centre as one of the ends of the minor axis

and the radius equal to the half of the major

axis. This method is used when we know only

major and minor axes of the ellipse.**9.If we know the major and minor axes of**

the ellipse, the first step of drawing the

ellipse, we draw the axes each other.

a) Parallel to

b) Perpendicular bisecting

c) Just touching

d) Coinciding**Answer: b**If we know the major and

Explanation:

minor axes of the ellipse, the first step of the

drawing the ellipse is to draw the major and

minor axes perpendicular bisecting each

other. The major and the minor axes are

perpendicular bisectors of each other.

**10.Loop of the thread method is the practical**

application of method.

a) Oblong method

b) Trammel method

c) Arcs of circles method

d) Concentric method**Answer: cExplanation:** Loop of the thread method is

the practical application of the arcs of circles

method. The lengths of the ends of the minor

axis are half of the length of the major axis.

In this method, a pin is inserted at the foci

point and the thread is tied to a pencil which

is used to draw the curve.

**CONSTRUCTION OF PARABOLA**

**1.Which of the following is incorrect about**

Parabola?

a) Eccentricity is less than 1

b) Mathematical equation is x^{2}= 4ay

c) Length of latus rectum is 4a

d) The distance from the focus to a vertex is

equal to the perpendicular distance from a

vertex to the directrix**Answer: a**The eccentricity is equal to one.

Explanation:

That is the ratio of a perpendicular distance

from point on curve to directrix is equal to

distance from point to focus. The eccentricity

is less than 1 for an ellipse, greater than one

for hyperbola, zero for a circle, one for a

parabola.**2.Which of the following constructions use**

parabolic curves?

a) Cooling towers

b) Water channels

c) Light reflectors

d) Man-holes**Answer: c**Arches, Bridges, sound

Explanation:

reflectors, light reflectors etc use parabolic

curves. Cooling towers, water channels use

Hyperbolic curves as their design. Arches,

bridges, dams, monuments, man-holes, glands

and stuffing boxes etc use elliptical curves.**3.The length of the latus rectum of the**

parabola y^{2}=ax is

a) 4a

b) a

c) a/4

d) 2a**Answer: b**Latus rectum is the line

Explanation:

perpendicular to axis and passing through

focus ends touching parabola. Length of latus

rectum of y^{2}=4ax, x^{2}=4ay is 4a; y^{2}=2ax, x^{2}=2ay is 2a; y^{2}=ax, x^{2}=ay is a.**4.Which of the following is not a parabola**

equation?

a) x^{2}= 4ay

b) y^{2}– 8ax = 0

c) x^{2}= by

d) x^{2}= 4ay^{2}**Answer: d**The remaining represents

Explanation:

different forms of parabola just by adjusting

them we can get general notation of parabola

but x^{2}= 4ay^{2}gives equation for hyperbola.

And x^{2}+ 4ay^{2}=1 gives equation for ellipse.**5.The parabola x2 = ay is symmetric about x-axis.**

a) True

b) False**Answer: b**From the given parabolic

Explanation:

equation x^{2}= ay we can easily say if we give

y values to that equation we get two values

for x so the given parabola is symmetric

about y-axis. If the equation is y^{2}= ax then it

is symmetric about x-axis.

**6.Steps are given to find the axis of a**

parabola. Arrange the steps.

i. Draw a perpendicular GH to EF which cuts

parabola.

ii. Draw AB and CD parallel chords to given

parabola at some distance apart from each

other.

iii. The perpendicular bisector of GH gives

axis of that parabola.

iv. Draw a line EF joining the midpoints lo

AB and CD.

a) i, ii, iii, iv

b) ii, iv, i, iii

c) iii, iv, i, ii

d) iv, i, ii, iii**Answer: bExplanation:** First we drawn the parallel

chords and then line joining the midpoints of

the previous lines which is parallel to axis so

we drawn the perpendicular to this line and

then perpendicular bisector gives the axis of

parabola.

**7.Steps are given to find focus for a parabola.**

Arrange the steps.

i. Draw a perpendicular bisector EF to BP,

Intersecting the axis at a point F.

ii. Then F is the focus of parabola.

iii. Mark any point P on the parabola and

draw a perpendicular PA to the axis.

iv. Mark a point B on the on the axis such that

BV = VA (V is vertex of parabola). Join B

and P.

a) i, ii, iii, iv

b) ii, iv, i, iii

c) iii, iv, i, ii

d) iv, i, ii, iii**Answer: c**Initially we took a parabola

Explanation:

with axis took any point on it drawn a

perpendicular to axis. And from the point

perpendicular meets the axis another point is

taken such that the vertex is equidistant from

before point and later point. Then from that

one to point on parabola a line is drawn and

perpendicular bisector for that line meets the

axis at focus.**8.Which of the following is not belonged to**

ellipse?

a) Latus rectum

b) Directrix

c) Major axis

d) Axis**Answer: c**Latus rectum is the line joining

Explanation:

one of the foci and perpendicular to the major

axis. Major axis and minor axis are in ellipse

but in parabola, only one focus and one axis

exist since eccentricity is equal to 1.

**CONSTRUCTION OF HYPERBOLA**

**1.Which of the following is Hyperbola**

equation?

a) y^{2}+ x^{2}/b^{2}= 1

b) x^{2}= 1ay

c)x^{2}/a^{2}– y^{2}/b^{2}= 1

d) x^{2}+ y^{2}= 1**Answer: c**The equation x

Explanation:^{2}+ y^{2}= 1 gives

a circle; if the x^{2}and y^{2}have same co-efficient then the equation gives circles. The

equation x^{2}= 1ay gives a parabola. The

equation y^{2}+ x^{2}/b^{2}= 1 gives an ellipse.**2.Which of the following constructions use**

hyperbolic curves?

a) Cooling towers

b) Dams

c) Bridges

d) Man-holes**Answer: a**Cooling towers, water channels

Explanation:

use Hyperbolic curves as their design.

Arches, Bridges, sound reflectors, light

reflectors etc., use parabolic curves. Arches,

bridges, dams, monuments, man-holes, glands

and stuffing boxes etc., use elliptical curves.

**3.The lines which touch the hyperbola at an**

infinite distance are

a) Axes

b) Tangents at vertex

c) Latus rectum

d) Asymptotes**Answer: d**Axis is a line passing through

Explanation:

the focuses of a hyperbola. The line which

passes through the focus and perpendicular to

the major axis is latus rectum. Tangent is the

line which touches the curve at only one

point.**4.Which of the following is the eccentricity**

for hyperbola?

a) 1

b) 3/2

c) 2/3

d) 1/2**Answer: b**The eccentricity for an ellipse

Explanation:

is always less than 1. The eccentricity is

always 1 for any parabola. The eccentricity is

always 0 for a circle. The eccentricity for a

hyperbola is always greater than 1.**5.If the asymptotes are perpendicular to each**

other then the hyperbola is called rectangular

hyperbola.

a) True

b) False**Answer: a**In ellipse there exist two axes

Explanation:

(major and minor) which are perpendicular to

each other, whose extremes have tangents

parallel them. There exist two conjugate axes

for ellipse and 1 for parabola and hyperbola.**6.A straight line parallel to asymptote**

intersects the hyperbola at only one point.

a) True

b) False**Answer: a**A straight line parallel to

Explanation:

asymptote intersects the hyperbola at only

one point. This says that the part of hyperbola

will lay in between the parallel lines through

outs its length after intersecting at one point.**7.Steps are given to locate the directrix of**

hyperbola when axis and foci are given.

Arrange the steps.

i. Draw a line joining A with the other Focus

F.

ii. Draw the bisector of angle FAF1, cutting

the axis at a point B.

iii. Perpendicular to axis at B gives directrix.

iv. From the first focus F1 draw a

perpendicular to touch hyperbola at A.

a) i, ii, iii, iv

b) ii, iv, i, iii

c) iii, iv, i, ii

d) iv, i, ii, iii**Answer: d**The directrix cut the axis at the

Explanation:

point of intersection of the angular bisector of

lines passing through the foci and any point

on a hyperbola. Just by knowing this we can

find the directrix just by drawing

perpendicular at that point to axis.**8.Steps are given to locate asymptotes of**

hyperbola if its axis and focus are given.

Arrange the steps.

i. Draw a perpendicular AB to axis at vertex.

ii. OG and OE are required asymptotes.

iii. With O midpoint of axis (centre) taking

radius as OF (F is focus) draw arcs cutting

AB at E, G.

iv.Join O, G and O, E.

a) i, iii, iv, ii

b) ii, iv, i, iii

c) iii, iv, i, ii

d) iv, i, ii, iii**Answer: b**Asymptotes pass through centre is the main point and then the asymptotes cut the directrix and perpendiculars at focus are known and simple. Next comes is where the asymptotes cuts the perpendiculars, it is at distance of centre to vertex and centre to focus respectively.

Explanation:

**9.The asymptotes of any hyperbola intersects at**

a) On the directrix

b) On the axis

c) At focus

d) Centre**Answer: d**The asymptotes intersect at

Explanation:

centre that is a midpoint of axis even for

conjugate axis it is valid. Along with the

hyperbola asymptotes are also symmetric

about both axes so they should meet at centre

only.

**CONSTRUCTION OF CYCLOIDAL CURVES**

**1.Is a curve generated by a**

point fixed to a circle, within or outside its

circumference, as the circle rolls along a

straight line.

a) Cycloid

b) Epicycloid

c) Epitrochoid

d) Trochoid**Answer: d**Cycloid form if generating

Explanation:

point is on the circumference of generating a

circle. Epicycloid represents generating circle

rolls on the directing circle. Epitrochoid is

that the generating point is within or outside

the generating circle but generating circle

rolls on directing circle.**2.Is a curve generated by a**

point on the circumference of a circle, which

rolls without slipping along another circle

outside it.

a) Trochoid

b) Epicycloid

c) Hypotrochoid

d) Involute**Answer: b**Trochoid is curve generated by

Explanation:

a point fixed to a circle, within or outside its

circumference, as the circle rolls along a

straight line. ‘Hypo’ represents the generating

circle is inside the directing circle.**3.Is a curve generated by a point**

on the circumference of a circle which rolls

without slipping on a straight line.

a) Trochoid

b) Epicycloid

c) Cycloid

d) Evolute**Answer: c**Trochoid is curve generated by

Explanation:

a point fixed to a circle, within or outside its

circumference, as the circle rolls along a

straight line. Cycloid is a curve generated by

a point on the circumference of a circle which

rolls along a straight line. ‘Epi’ represents the

directing path is a circle.**4.When the circle rolls along another circle**

inside it, the curve is called a

a) Epicycloid

b) Cycloid

c) Trochoid

d) Hypocycloid**Answer: d**Cycloid is a curve generated by

Explanation:

a point on the circumference of a circle which

rolls along a straight line. ‘Epi’ represents the

directing path is a circle. Trochoid is a curve

generated by a point fixed to a circle, within

or outside its circumference, as the circle rolls

along a straight line. ‘Hypo’ represents the

generating circle is inside the directing circle.

**6.Match the following****Generating point is – i. Inferior within the circumference trochoid**

**Generating point is on the circumference of circle and – ii. Epicycloid generating circle rolls onstraight line.**

**The circumference of circle and generating circle rolls on straight line – iii. Cycloid**

**Generating point is on the circumference of circle **

**and generating circle rolls – iv. Superiortrochoid**

a) 1, i; 2, iii; 3, iv; 4, ii

b) 1, ii; 2, iii; 3, i; 4, iv

c) 1, ii; 2, iv; 3, iii; 4, i

d) 1, iv; 2, iii; 3, ii; 4, i**Answer: a**

**Explanation:** Trochoid is curve generated by

a point fixed to a circle, within or outside its

circumference, as the circle rolls along a

straight line. Inferior or superior depends on

whether the generating point in within or

outside the generating circle. If directing path

is straight line then the curve is cycloid.

**7.Steps are given to find the normal and**

tangent for a cycloid. Arrange the steps if C is

the centre for generating circle and PA is the

directing line. N is the point on cycloid.**i. Through M, draw a line MO perpendicular**

to the directing line PA and cutting at O.

ii. With centre N and radius equal to radius of

generating circle, draw an arc cutting locus of

C at M.

**iii. Draw a perpendicular to ON at N which is tangent.**

**iv. Draw a line joining O and N which isnormal.**

a) iii, i, iv, ii

b) ii, i, iv, iii

c) iv, ii, i, iii

d) i, iv, iii, ii**Answer: bExplanation:** The normal at any point on a cycloidal curve will pass through the corresponding point of contact between the generating circle and the directing line. So with help of locus of centre of generating circle we found the normal and the tangent.

**8.Steps are given to find the normal and**

tangent to an epicycloid. Arrange the steps if

C is the centre for generating circle and O is

the centre of directing cycle. N is the point on

epicycloid.**i. Draw a line through O and D cutting**

directing circle at M.

ii. Draw perpendicular to MN at N. We get

tangent.

iii. With centre N and radius equal to radius

of generating circle, draw an arc cutting the

locus of C at D.

iv. Draw a line joining M and N which is

normal.

a) iii, i, iv, ii

b) ii, i, iv, iii

c) iv, ii, i, iii

d) i, iv, iii, ii**Answer: a**The normal at any point on an

Explanation:

epicycloidal curve will pass through the

corresponding point of contact between the

generating circle and the directing circle. And

also with help of locus of centre of generating

circle we found the normal and the tangent.**9.The generating circle will be inside the**

directing circle for

a) Cycloid

b) Inferior trochoid

c) Inferior epitrochoid

d) Hypocycloid**Answer: d**The generating circle will be

Explanation:

inside the directing circle for hypocycloid or

hypotrochoid. Trochoid is a curve generated

by a point fixed to a circle, within or outside

its circumference, as the circle rolls along a

straight line or over circle if not represented

with hypo as a prefix.**10.The generating point is outside the**

generating circle for

a) Cycloid

b) Superior Trochoid

c) Inferior Trochoid

d) Epicycloid**Answer: b**If the generating point is on the

Explanation:

circumference of generating circle then the

curve formed may be cycloids or

hypocycloids. Trochoid is a curve generated

by a point fixed to a circle, within or outside

its circumference, as the circle rolls along a

straight line or a circle. But here given is

outside so it is superior trochoid.

**CONSTRUCTION OF INVOLUTE**

**1.Mathematical equation for Involute is**

a) x = a cos3θ

b) x = r cosθ + r θ sinθ

c) x = (a+b)cosθ – a cos(a+b⁄a θ)

d) y = a(1-cosθ)**Answer: b**x= a cos3 Ɵ is equation forhypocycloid, x= (a+ b) cosƟ – a cos ((a+b)/aƟ) is equation for epicycloid, y= a (1-cosƟ) is equation for cycloid and x = r cosƟ r Ɵ sinƟ is equation for Involute.

Explanation:

**2.Steps are given to draw involute of given**

circle. Arrange the steps f C is the centre of

circle and P be the end of the thread (starting

point).

i. Draw a line PQ, tangent to the circle and

equal to the circumference of the circle.

ii. Draw the involute through the points P1,

P2, P3 ……….etc.

iii. Divide PQ and the circle into 12 equal

parts.

iv. Draw tangents at points 1, 2, 3 etc. and

mark on them points P1, P2, P3 etc. such that

1P1 =P1l, 2P2 = P2l, 3P3= P3l etc.

a) ii, i, iv, iii

b) iii, i , iv, ii

c) i, iii, iv, ii

d) iv, iii, i, ii**Answer: cExplanation:** Involute is a curve which is

formed by the thread which is yet complete a

single wound around a circular object so thus

the thread having length equal to the

circumference of the circular object. And the

involute curve follows only the thread is kept

straight while wounding.

**3.Steps are given to draw tangent and normal**

to the involute of a circle (center is C) at a

point N on it. Arrange the steps.

i. With CN as diameter describe a semi-circle

cutting the circle at M.

ii. Draw a line joining C and N.

iii. Draw a line perpendicular to NM and

passing through N which is tangent.

iv. Draw a line through N and M. This line is

normal.

a) ii, i, iv, iii

b) iii, i , iv, ii

c) i, iii, iv, ii

d) iv, iii, i, ii**Answer: a**The normal to an involute of a

Explanation:

circle is tangent to that circle. So simply by

finding the appreciable tangent of circle

passing through the point given on involute

gives the normal and then by drawing

perpendicular we can find the tangent to

involute.**4.Steps given are to draw an involute of a**

given square ABCD. Arrange the steps.

i. With B as centre and radius BP1 (BA+ AD)

draw an arc to cut the line CB-produced at

P2.

ii. The curve thus obtained is the involute of

the square.

iii. With centre A and radius AD, draw an arc

to cut the line BA-produced at a point P1.

iv. Similarly, with centres C and D and radii

CP2 and DP3 respectively, draw arcs to cut

DC-produced at P3 and AD-produced at P4.

a) ii, i, iv, iii

b) iii, i , iv, ii

c) i, iii, iv, ii

d) iv, iii, i, ii**Answer: b**It is easy to draw involutes to

Explanation:

polygons. First, we have to point the initial

point and then extending the sides. Then

cutting the extended lines with cumulative

radiuses of length of sides gives the points on

involute and then joining them gives involute.**5.Steps given are to draw an involute of a**

given triangle ABC. Arrange the steps.

i. With C as centre and radius C1 draw arc

cutting AC-extended at 2.

ii. With A as center and radius A2 draw an

arc cutting BA- extended at 3 completing

involute.

iii. B as centre with radius AB draw an arc

cutting the BC- extended at 1.

iv. Draw the given triangle with corners A, B, C.

a) ii, i, iv, iii

b) iii, i , iv, ii

c) i, iii, iv, ii

d) iv, iii, i, ii**Answer: d**It will take few simple steps to

Explanation:

draw involute for a triangle since it has only 3

sides. First, we have to point the initial point

and then extending the sides. Then cutting the

extended lines with cumulative radiuses of

length of sides gives the points on involute

and then joining them gives involute.**6.Steps given are to draw an involute of a**

given pentagon ABCDE. Arrange the steps.

i. B as centre and radius AB, draw an arc

cutting BC –extended at 1.

ii. The curve thus obtained is the involute of

the pentagon.

iii. C as centre and radius C1, draw an arc

cutting CD extended at 2.

iv. Similarly, D, E, A as centres and radiusD2, E3, A4, draw arcs cutting DE, EA, AB at 3, 4, 5 respectively.

a) ii, i, iv, iii

b) iii, i , iv, ii

c) i, iii, iv, ii

d) iv, iii, i, ii**Answer: cExplanation:** It is easy to draw involutes to

polygons. First, we have to point the initial

point and then extending the sides. Then

cutting the extended lines with cumulative

radiuses of length of sides gives the points on

involute and then joining them gives involute.

**7.For inferior trochoid or inferior epitrochoid**

the curve touches the directing line or

directing circle.

a) True

b) False**Answer: b**Since in the inferior trochoids

Explanation:

the generating point is inside the generating

circle the path will be at a distance from

directing line or circle even if the generating

circle is inside or outside the directing circle.**8.‘Hypo’ as prefix to cycloids give that the**

generating circle is inside the directing circle.

a) True

b) False**Answer: a**‘Hypo’ represents the

Explanation:

generating circle is inside the directing circle.

‘Epi’ represents the directing path is a circle.

Trochoid represents the generating point is

not on the circumference of generating a

circle.

**CONSTRUCTION OF SPIRAL**

**1.Which of the following represents an**

Archemedian spiral?

a) Tornado

b) Cyclone

c) Mosquito coil

d) Fibonacci series**Answer: c**Archemedian spiral is a curve

Explanation:

traced out by a point moving in such a way

that its movement towards or away from the

pole is uniform with the increase of the

vectorial angle from the starting line. It is

generally used for teeth profiles of helical

gears etc.**2.Steps are given to draw normal and tangent**

to an archemedian curve. Arrange the steps, if

O is the center of curve and N is point on it.

i. Through N, draw a line ST perpendicular to

NM. ST is the tangent to the spiral.

ii. Draw a line OM equal in length to the

constant of the curve and perpendicular to

NO.

iii. Draw the line NM which is normal to the

spiral.

iv. Draw a line passing through the N and O

which is radius vector.

a) ii, iv, i, iii

b) i, iv, iii, ii

c) iv, ii, iii, i

d) iii, i, iv, ii**Answer: c**The normal to an archemedian

Explanation:

spiral at any point is the hypotenuse of the

right angled triangle having the other two

sides equal in length to the radius vector at

that point and the constant of the curve

respectively.**3.Which of the following does not represents**

an Archemedian spiral?

a) Coils in heater

b) Tendrils

c) Spring

d) Cyclone**Answer: d**Tendrils are a slender thread-like structures of a climbing plant, often growing in a spiral form. For cyclones the moving point won’t have constant velocity. The archemedian spirals have a constant increase in the length of a moving point. Spring is a helix.

Explanation:**4.Logarithmic spiral is also called**

Equiangular spiral.

a) True

b) False**Answer: a**The logarithmic spiral is also

Explanation:

known as equiangular spiral because of its

property that the angle which the tangent at

any point on the curve makes with the radius

vector at that point is constant. The values of

vectorial angles are in arithmetical

progression.**In logarithmic Spiral, the radius vectors are**

in arithmetical progression.

a) True

b) False**Answer: b**In the logarithmic Spiral, the

Explanation:

values of vectorial angles are in arithmetical

progression and radius vectors are in the

geometrical progression that is the lengths of

consecutive radius vectors enclosing equal

angles are always constant.**The mosquito coil we generally see in**

house hold purposes and heating coils in

electrical heater etc are generally which

spiral.

a) Logarithmic spiral

b) Equiangular spiral

c) Fibonacci spiral

d) Archemedian spiral**Answer: d**Archemedian spiral is a curve

Explanation:

traced out by a point moving in such a way

that its movement towards or away from the

pole is uniform with the increase of the

vectorial angle from the starting line. The use

of this curve is made in teeth profiles of

helical gears, profiles of cam etc.

**BASICS OF CONIC SECTIONS – 1**

**1.The sections cut by a plane on a right**

circular cone are called as

a) Parabolic sections

b) Conic sections

c) Elliptical sections

d) Hyperbolic sections**Answer: b**The sections cut by a plane on a

Explanation:

right circular cone are called as conic sections

or conics. The plane cuts the cone on

different angles with respect to the axis of the

cone to produce different conic sections.**2.Which of the following is a conic section?**

a) Circle

b) Rectangle

c) Triangle

d) Square**Answer: a**Circle is a conic section. When

Explanation:

the plane cuts the right circular cone at right

angles with the axis of the cone, the shape

obtained is called as a circle. If the angle is

oblique we get the other parts of the conic

sections.**3.In conics, the is revolving to form**

two anti-parallel cones joined at the apex.

a) Ellipse

b) Circle

c) Generator

d) Parabola**Answer: c**In conics, the generator is

Explanation:

revolving to form two anti-parallel cones

joined at the apex. The plane is then made to

cut these cones and we get different conic

sections. If we cut at right angles with respect

to the axis of the cone we get a circle.**4.While cutting, if the plane is at an angle**

and it cuts all the generators, then the conic

formed is called as

a) Circle

b) Ellipse

c) Parabola

d) Hyperbola**Answer: b**If the plane cuts all the

Explanation:

generators and is at an angle to the axis of the

cone, then the resulting conic section is called

as an ellipse. If the cutting angle was right

angle and the plane cuts all the generators

then the conic formed would be circle.

**5.If the plane cuts at an angle to the axis but**

does not cut all the generators then what is

the name of the conics formed?

a) Ellipse

b) Hyperbola

c) Circle

d) Parabola**Answer: dExplanation:** If the plane cuts at an angle

with respect to the axis and does not cut all

the generators then the conics formed is a

parabola. If the plane cuts all the generators

then the conic section formed is called as

ellipse.

**6.When the plane cuts the cone at angle**

parallel to the axis of the cone, then is

formed.

a) Hyperbola

b) Parabola

c) Circle

d) Ellipse**Answer: a**When the plane cuts the cone at

Explanation:

an angle parallel to the axis of the cone, then

the resulting conic section is called as a

hyperbola. If the plane cuts the cone at an

angle with respect to the axis of the cone then

the resulting conic sections are called as

ellipse and parabola.**7.Which of the following is not a conic**

section?

a) Apex

b) Hyperbola

c) Ellipse

d) Parabola**Answer: a**Conic sections are formed

Explanation:

when a plane cuts through the cone at an

angle with respect to the axis of the cone. If

the angle is right angle then the conics is a

circle, if the angle is oblique then the

resulting conics are parabola and ellipse.**8.The locus of point moving in a plane such**

that the distance between a fixed point and a

fixed straight line is constant is called as

a) Conic

b) Rectangle

c) Square

d) Polygon**Answer: a**The locus of a point moving in

Explanation:

a plane such that the distance between a fixed

point and a fixed straight line is always

constant. The fixed straight line is called as

directrix and the fixed point is called as the

focus.**9.The ratio of the distance from the focus to**

the distance from the directrix is called as

eccentricity.

a) True

b) False**Answer: a**The ratio of the distance from

Explanation:

the focus to the distance from the directrix is

called eccentricity. It is denoted as e. The

value of eccentricity can give information

regarding which type of conics it is.**10.Which of the following conics has an**

eccentricity of unity?

a) Circle

b) Parabola

c) Hyperbola

d) Ellipse**Answer: b**Eccentricity is defined as the

Explanation:

ratio of the distance from the focus to the

distance from the directrix. It is denoted as e.

The value of eccentricity can give

information regarding which type of conics it

is. The eccentricity of a parabola is the unity

that is 1.**Which of the following has an**

eccentricity less than one?

a) Circle

b) Parabola

c) Hyperbola

d) Ellipse**Answer: dExplanation:** Eccentricity is defined as the

ratio of the distance from the focus to the

distance from the directrix. It is denoted as e.

The value of eccentricity can give

information regarding which type of conics it

is. The eccentricity of an ellipse is less than

one.

**12.If the distance from the focus is 10 units**

and the distance from the directrix is 30 units,

then what is the eccentricity?

a) 0.3333

b) 0.8333

c) 1.6667

d) 0.0333**Answer: a**Eccentricity is defined as the

Explanation:

ratio of the distance from the focus to the

distance from the directrix. Hence from the

formula of eccentricity, e = 10 ÷ 30 = 0.3333.

Since the value of eccentricity is less than one

the conic is an ellipse.**13.If the value of eccentricity is 12, then**

what is the name of the conic?

a) Ellipse

b) Hyperbola

c) Parabola

d) Circle**Answer: b**Eccentricity is defined as the

Explanation:

ration of the distance from the focus to the

distance from the directrix. It is denoted as e.

If the value of eccentricity is greater than

unity then the conic section is called as a

hyperbola.**14.If the distance from the focus is 3 units**

and the distance from the directrix is 3 units,

then how much is the eccentricity?

a) Infinity

b) Zero

c) Unity

d) Less than one**Answer: c**Eccentricity is defined as the

Explanation:

ration of the distance from the focus to the

distance from the directrix and it is denoted as

e. Hence from the definition, e = 3 ÷ 3 = 1.

Hence the value of eccentricity is equal to

unity.**15.If the distance from the focus is 2 mm and**

the distance from the directrix is 0.5 mm then

what is the name of the conic section?

a) Circle

b) Ellipse

c) Parabola

d) Hyperbola**Answer: d**The eccentricity is defined as

Explanation:

the ratio of the distance from the focus to the

distance from the directrix. It is denoted as e.

If the value of the eccentricity is greater than

unity then the conic section is called as a

hyperbola.

**BASICS OF CONIC SECTIONS – 2**

**1.Which of the following is a conic section?**

a) Apex

b) Circle

c) Rectangle

d) Square**Answer: b**Conic sections are formed

Explanation:

when a plane cuts through the cone at an

angle with respect to the axis of the cone. If

the angle is right angle then the conics is a

circle, if the angle is oblique then the

resulting conics are parabola and ellipse.**2.Which of the following has an eccentricity**

more than unity?

a) Parabola

b) Circle

c) Hyperbola

d) Ellipse**Answer: cExplanation:** Eccentricity is defined as the

ratio of the distance from the focus to the

distance from the directrix. It is denoted as e.

The value of eccentricity can give

information regarding which type of conics it

is. The eccentricity of a hyperbola is more

than one.

**3.If the distance from the focus is 10 units**

and the distance from the directrix is 30 units,

then what is the name of the conic?

a) Circle

b) Parabola

c) Hyperbola

d) Ellipse**Answer: d**Eccentricity is defined as the

Explanation:

ratio of the distance from the focus to the

distance from the directrix. Hence from the

formula of eccentricity, e = 10 ÷ 30 = 0.3333.

Since the value of eccentricity is less than one

the conic is an ellipse.**4.If the distance from the focus is 2 mm and**

the distance from the directrix is 0.5 mm then

what is the value of eccentricity?

a) 0.4

b) 4

c) 0.04

d) 40**Answer: b**Eccentricity is defined as the

Explanation:

ratio of the distance from the focus to the

distance from the directrix and it is denoted

by e. Therefore, by definition, e = 2 ÷ 0.5 = 4.

Hence the conic section is called as

hyperbola.**5.If the distance from the focus is 3 units and**

the distance from the directrix is 3 units, then

what is the name of the conic section?

a) Ellipse

b) Hyperbola

c) Circle

d) Parabola**Answer: d**Eccentricity is defined as the

Explanation:

ratio of the distance from the focus to the

distance from the directrix and it is denoted

by e. Therefore, by definition, e = 3 ÷ 3 = 1.

Hence the conic section is called as a

parabola.**6.If the distance from the directrix is 5 units**

and the distance from the focus is 3 units then

what is the name of the conic section?

a) Ellipse

b) Parabola

c) Hyperbola

d) Circle**Answer: a**Eccentricity is defined as the

Explanation:

ratio of the distance from the focus to the

distance from the directrix and it is denoted

by e. Hence, by definition, e = 3 ÷ 5 = 0.6.

Hence the conic section is called an ellipse.**7.If the distance from a fixed point is greater**

than the distance from a fixed straight line

then what is the name of the conic section?

a) Parabola

b) Circle

c) Hyperbola

d) Ellipse**Answer: c**The fixed point is called as

Explanation:

focus and the fixed straight line is called as

directrix. Eccentricity is defined as the ratio

of the distance from the focus to the distance

from the directrix and it is denoted by e. If e

is greater than one then the conic section is

called as a hyperbola.**8.If the distance from a fixed straight line is**

equal to the distance from a fixed point then

what is the name of the conic section?

b) Parabola

c) Hyperbola

d) Circle**Answer: bExplanation:** The fixed straight line is called

as directrix and the fixed point is called as a

focus. Eccentricity is defined as the ratio of

the distance from the focus to the distance

from the directrix and it is denoted by e.

Eccentricity of a parabola is unity.

**9.If the distance from the directrix is greater**

than the distance from the focus then what is

the value of eccentricity?

a) Unity

b) Less than one

c) Greater than one

d) Zero**Answer: b**Eccentricity is defined as the

Explanation:

ratio of the distance from the focus to the

distance from the directrix and it is denoted

by e. Therefore, by definition the value of

eccentricity is less than one hence the conic

section is an ellipse.**10.If the distance from the directrix is 5 units**

and the distance from the focus is 3 units then

what is the value of eccentricity?

a) 1.667

b) 0.833

c) 0.60

d) 0.667**Answer: c**Eccentricity is defined as the

Explanation:

ratio of the distance from the focus to the

distance from the directrix and it is denoted

by e. Therefore, by definition, e = 3 ÷ 5 = 0.6.

Hence the conic section is called an ellipse.**11.If the distance from a fixed straight line is**

5mm and the distance from a fixed point is

14mm then what is the name of the conic

section?

a) Hyperbola

b) Parabola

c) Ellipse

d) Circle**Answer: a**The fixed straight line is called

Explanation:

directrix and the fixed point is called as a

focus. Eccentricity is defined as the ratio of

the distance from the focus to the distance

from the directrix and it is denoted by e.

Hence from definition e = 14 ÷ 5 = 2.8. The

eccentricity of a hyperbola is greater than

one.**12.If the distance from the directrix is greater**

than the distance from the focus then what is

the name of the conic section?

a) Hyperbola

b) Parabola

c) Ellipse

d) Circle**Answer: c**Eccentricity is defined as the

Explanation:

ratio of the distance from the focus to the

distance from the directrix and it is denoted

by e. Therefore, by definition the value of

eccentricity is less than one hence the conic

section is an ellipse.**13.If the distance from a fixed straight line is**

equal to the distance from a fixed point then

what is the value of eccentricity?

a) Unity

b) Greater than one

c) Infinity

d) Zero**Answer: a**The fixed straight line is called

Explanation:

as directrix and the fixed point is called as a

focus. Eccentricity is defined as the ratio of

the distance from the focus to the distance

from the directrix and it is denoted by e.

Hence from definition e = x ÷ x = 1.

**14.If the distance from a fixed point is**

greater than the distance from a fixed straight

line then what is the value of eccentricity?

a) Unity

b) Infinity

c) Zero

d) Greater than one**Answer: d**The fixed point is called as

Explanation:

focus and the fixed straight line is called as

directrix. Eccentricity is defined as the ratio

of the distance from the focus to the distance

from the directrix and it is denoted by e.

Hence from the definition, the value of

eccentricity is greater than one.**15.If the distance from a fixed straight line is**

5mm and the distance from a fixed point is

14mm then what is the value of eccentricity?

a) 0.357

b) 3.57

c) 2.8

d) 0.28**Answer: c**The fixed straight line is called

Explanation:

as directrix and the fixed point is called as a

focus. Eccentricity is defined as the ratio of

the distance from the focus to the distance

from the directrix and it is denoted by e.

Hence from definition e = 14 ÷ 5 = 2.8.

**BASICS OF CONIC SECTIONS – 3**

**1.Choose the correct option.**

(a)Eccentricity= Distance of the point from the focus/ Distance of the point from the vertex

(b)Eccentricity= Distance of the point from the focus/ Distance of the point from the directrix

(c)Eccentricity= Distance of the point from the directrix / Distance of the point from the focus

(d)Eccentricity= Distance of the point from the latus rectum/ Distance of the point from the focus

**Answer: bExplanation:** The point where the extension

of major axis meets the curve is called vertex.

The conic is defined as the locus of a point in

such a way that the ratio of its distance from a

fixed point and a fixed straight line is always

constant. The ratio gives the eccentricity. The

fixed point is called the focus and the fixed

line is called directrix.

2. **2.Match the following.**

**A. E < 1 i. Rectangular hyperbola **

**B. E = 1 ii. Hyperbola **

**C. E > 1 iii. Ellipse **

**D. E > 1 iv. Parabola **

a) A, i; B, ii; C, iii; D, iv

b) A, ii; B, iii; C, iv; D, i

c) A, iii; B, iv; C, ii; D, i

d) A, iv; B, iii; C, ii; D, i

**Answer: c **

**Explanation:** The conic is defined as the locus of a point in such a way that the ratio of its distance from a fixed point and a fixed straight line is always constant. The fixed point is called the focus and the fixed line is called directrix. The change in ratio as given above results in different curves.

**3.A plane is parallel to a base of regular cone**

and cuts at the middle. The cross-section is

a) Circle

b) Parabola

c) Hyperbola

d) Ellipse**Answer: aExplanation:** A cone is formed by reducing

the cross-section of a circle the point. So

there exist circles along the cone parallel to

the base. Since the given plane is parallel to

the base of the regular cone. The crosssection will be circle

**4.The cross-section is a when a**

plane is inclined to the axis and cuts all the

generators of a regular cone.

a) Rectangular Hyperbola

b) Hyperbola

c) Circle

d) Ellipse**Answer: d**A cone is a solid or hollow

Explanation:

object which tapers from a circular base to a

point. Here given an inclined plane which

cuts all the generators of a regular cone. So

the cross-section will definitely ellipse.**5.The curve formed when eccentricity is**

equal to one is

a) Parabola

b) Circle

c) Semi-circle

d) Hyperbola**Answer: a**The answer is parabola. Circle

Explanation:

has an eccentricity of zero and semi circle is

part of circle and hyper eccentricity is greater

than one.

**6.The cross-section gives a when the cutting plane is parallel to axis of cone.**

a) Parabola

b) Hyperbola

c) Circle

d) Ellipse**Answer: b**If the cutting plane makes angle

Explanation:

less than exterior angle of the cone the crosssection gives a ellipse. If the cutting plane

makes angle greater than the exterior angle of

the cone the cross- section may be parabola or

hyperbola.

**7.A plane cuts the cylinder the plane is not**

parallel to the base and cuts all the generators.

The Cross-section is

a) Circle

b) Ellipse

c) Parabola

d) Hyperbola**Answer: b**Given is a plane which is

Explanation:

inclined but cutting all the generators so it

will be an ellipse. Cutting of all generators

gives us information that the cross-section

will be closed curve and not parabola or

hyperbola. Circle will form only if plane is

parallel to the base.**8.A plane cuts the cylinder and the plane is**

parallel to the base and cuts all the generators.

The Cross-section is

a) Circle

b) Ellipse

c) Parabola

d) Rectangular hyperbola**Answer: a**The plane which is parallel to base will definitely cut the cone at all generators. Here additional information also given that the plane is parallel to base so the cross-section will be circle.

Explanation:

**9.The curve which has eccentricity zero is**

a) Parabola

b) Ellipse

c) Hyperbola

d) Circle**Answer: d**The eccentricity is the ratio of a

Explanation:

distance from a point on the curve to focus

and to distance from the point to directrix.

For parabola it is 1 and for ellipse it is less

than 1 and for hyperbola it is greater than 1.

And for circle it is zero.**10.Rectangular hyperbola is one of the**

hyperbola but the asymptotes are

perpendicular in case of rectangular

hyperbola.

a) True

b) False**Answer: a**Asymptotes are the tangents

Explanation:

which meet the curve hyperbola at infinite

distance. If the asymptotes are perpendicular

to each other then hyperbola takes the name

of a rectangular hyperbola.